Theory of Statistics - George Mason University

0.1 Measure, Integration, and Functional Analysis 729
• If f(x) is continuous on [a, b], then it is Riemann integrable over [a, b].
• If f(x) is monotone (increasing or decreasing) on [a, b], then it is Riemann
integrable over [a, b]. (Notice that the function may not be continuous.)
• If f(x) is of bounded variation on [a, b], then it is Riemann integrable over
[a, b].
0.1.6.10 The Riemann-Stieltjes Integral
The Riemann-Stieltjes integral is a generalization of the Riemann integral in
which dx is replaced by dg(x) and the interval lengths are replaced by changes
in g(x). We write it as
rs =
b
a
f(x)dg(x). (0.1.57)
To define the Riemann-Stieltjes integral we will handle the partitions
slightly differently from how they were used in equation (0.1.55) for the
Riemann integral. (Either way could be used for either integral, however.
This is different from integrals with respect to stochastic differentials, where
the endpoints matter; see Section 0.2.2.) Form a partition of [a, b], call it
P = (a = x0 < x1 < · · · < xn = b), and let ∆gi = g(xi) − g(xi−1). We now
consider the sup and inf of f within each interval of the partition and the inf
and sup of sums of over all partitions:
and
inf
P
sup
P
n
sup f(x)∆gi
x∈[xi,xi−1]
i=1
n
inf
x∈[xi,xi−1] f(x)∆gi.
i=1
If these are equal, then the Riemann-Stieltjes integral is defined as their common
value:
b
a
f(x)dg(x) = inf
P
n
sup
i=1
x∈[xi,xi−1]
f(x)∆gi = sup
P
n
inf
x∈[xi,xi−1] f(x)∆gi.
(0.1.58)
There is a simple connection with Riemann-Stieltjes integral and the Riemann
integral whenever g ′ (x) exists and is continuous.
Theorem 0.1.27
Suppose that Riemann-Stieltjes integral b
f(x)dg(x) exists and suppose the
a
derivative of g, g ′ (x) exists and is continuous on [a, b]; then
b
a
f(x)dg(x) =
b
a
i=1
f(x)g ′ (x)dx.
Theory of Statistics c○2000–2013 James E. Gentle